The paper revisits the Elitzur-Vaidman bomb-testing thought experiment, in which a single photon in a Mach-Zehnder interferometer can reveal that a photon-triggered bomb sits in one arm without being absorbed by it. It derives the theoretical outcome probabilities for 50/50 beam splitters (bomb triggered 1/2, each detector 1/4) and reviews proposed and previously reported quantum-Zeno variants in which N repeated looks raise the success probability to [cos^2(pi/2N)]^N. The authors' own work is a qualitative HeNe-laser Mach-Zehnder demonstration and quantum-circuit simulations run through Qiski

The Elitzur–Vaidman bomb test

No photons yet

Can you check that a bomb works without setting it off? Its trigger fires the moment it absorbs a single photon, and normally the only way to test a trigger is to shine light on it. Here a single photon is split between two paths that meet again. At detector D2 the two halves cancel, so D2 never fires. Put a live bomb in the upper path, and D2 starts to click. Each click proves the bomb is live, yet the photon never touched it.

Simulated single photons, one at a time, sampled from the probabilities derived in the paper (Sec. 2). The bar ticks mark the prediction. The two dials show each path’s contribution at each detector: they add at D1 and cancel at D2.

The puzzle: to look is to touch

Every measurement is an interaction. To learn where something is, or what state it is in, you send in a probe, usually light, and read what comes back. Light bounces off the object or is absorbed by it, and that contact is what carries the information to you. Most of the time the disturbance is too small to matter.

Elitzur and Vaidman pushed this to the extreme. Imagine bombs that are live , with a trigger so sensitive that absorbing a single photon sets them off, mixed with duds whose trigger is broken, so light passes straight through. As the paper puts it, “classically, it is impossible to know if the bomb works without disturbing the system.” Try it with ordinary light:

Whatever you do, the mint number stays at zero. Can you ever learn that a trigger works without a single photon reaching it?

Superposition. In an interferometer, one photon travels both paths at once until it is detected.

Interference. The two paths are tuned to cancel at detector D2, so with nothing in the way D2 stays dark.

A possible measurement. A live bomb in one path could absorb the photon, which would reveal its path. That possibility alone destroys the cancellation, so D2 can click on runs where the bomb was never touched.

That is the interferometer at the top of the page. First, a closer look at step 01: what “both paths at once” means for a single photon.

One photon, no definite path

It is tempting to picture the photon as a tiny ball that goes one way or the other at the first beam splitter. Quantum mechanics says otherwise. Inside the interferometer the photon does not take path a or path b: it is in a superposition of both. In the paper’s words, “the single photon exists simultaneously in both paths A and B until a measurement is made.” Each path holds one of the photon’s possible states, and the two possibilities meet again at the second splitter, where they interfere.

You can check whether the photon has a definite path by looking for it. Switch on the markers below. They are idealized detectors that record which path the photon is in. Each photon is now found in one path, but which one is pure chance, 50/50 every time. Nothing about the photon or the setup predicts it. Guess before each photon and see how you do. You can watch the photon travel from the source to the first splitter, where it still has a single, definite path. There it stops having one. Past that point the photon is no longer drawn, because no single path for it exists to draw: all you get are clicks, at D1, at D2 or at a marker.

Compare the detectors. Unwatched, every photon reaches D1, because the two possibilities cancel at D2. Watched, the record of the path is a random string, and D1 and D2 now click equally often: the interference is gone. As the paper puts it, “obtaining information about the photon’s path destroys the interference pattern.” A live bomb in one arm does exactly this kind of looking, and that is the whole trick of the bomb test.

Simulated single photons, sampled one at a time from the standard probabilities for 50/50 beam splitters. The path markers are an idealized illustration; in the paper the path is “measured” by an object that absorbs the photon, which also destroys the interference (Sec. 2).

Sort the shelf without blowing it up

Elitzur and Vaidman framed their puzzle as quality control. With ordinary light, every live bomb you confirm, you destroy. Now you have the interferometer instead. Pick a sealed crate, load it into path a, and fire photons one at a time.

D2 click proves the bomb is live and intact. D2 can only fire if path a is blocked.

Boom also proves it was live, but now it is gone.

D1 click is inconclusive. A dud always gives D1, and a live bomb gives D1 a quarter of the time.

Play it a thousand times

The protocol: fire until D2 clicks (certify) or the bomb explodes. After six D1 clicks in a row, call it a dud. The paper’s measure of success is η = P(D2) / (P(boom) + P(D2)). With 50/50 beam splitters that is ¼ / (½ + ¼) = 1/3 .

1,000 live bombs and 1,000 duds, sampled in your browser.

A simple bench setup to see the interference

To see the effect for themselves, the authors built a Mach–Zehnder interferometer from two 50/50 beam splitters and a helium–neon laser at 632.8 nm. With both arms open, one output shows a pattern of rings and the other shows the complementary pattern, bright where the first is dark. When an object blocks one arm, the rings disappear and the previously dark region lights up.

A continuous laser beam, not single photons. As the authors note, this bench shows the interference qualitatively and cannot by itself test the single-photon prediction.

The ring pattern is an illustrative model of what the paper describes (Figs. 5–6), not the authors’ photographs.

Two qubits play photon and bomb

The authors also turned the experiment into a quantum circuit and ran it through IBM’s cloud quantum computing service (Qiskit Runtime). One qubit is the photon: its value, 0 or 1, marks the path, and at the end the detector. A second qubit is the bomb. The Hadamard gate H acts as a 50/50 beam splitter. A controlled-NOT gate ( CNOT ) flips the bomb qubit when the photon takes path 1. That is the bomb noticing the photon.

Without the bomb gate, the second H undoes the first and every shot reads D1: that is interference. With the gate in, the four outcomes each appear about a quarter of the time, and the paper’s IBM run shows four near-equal bars of about 0.25 (Fig. 7). The outcome in mint, dark port with the bomb quiet , is the interaction-free detection, in 25% of runs. The shots on this page are sampled in your browser from the same circuit.

Tune the beam splitter: the one-half ceiling

Is one in three the best you can do? The paper gives the general answer. If the first beam splitter sends a fraction R (its reflectivity) toward the bomb, a live bomb explodes with probability R, and D2 clicks with probability R(1−R). Repeating on every D1, the share of live bombs certified intact is η = (1−R)/(2−R) .

A gentler splitter (small R) wastes fewer bombs but needs many more photons per answer, and η creeps toward 50% without ever passing it. To break the ceiling, you have to look more gently, many times. That is what a chain of interferometers can do.

A bomb in every arm

The way past ½ is the quantum Zeno effect : frequent observations can delay or even halt a quantum system’s evolution. The paper describes a version proposed by Kwiat and colleagues in which a single photon passes through a Mach–Zehnder interferometer again and again; its Fig. 4 unrolls the passes into a chain of linked interferometers. The beam splitters are very unequal: at each one the photon keeps the lower path with probability cos²(π/2N) , and only a sliver of its amplitude moves to the upper path. With no bomb, the slivers add up in step and the photon drifts from the lower-left half of the system to the upper-right half, ending at D1.

Now put a live bomb in the upper arm of every interferometer, as drawn in the paper. Every interferometer asks “is the photon here?” Almost always the answer is no, which puts the photon back on the lower path, and the drift starts again from zero. A photon that survives all N splitters exits at D2 with probability [cos²(π/2N)]N , which rises toward 1 as you add splitters. With two splitters this is the ordinary bomb test: D2 one time in four.

The three outcomes are the paper’s: the photon is absorbed and the bomb goes off; D1 fires, which is what an empty path gives (with the bombs in place it becomes rare); or D2 fires, meaning the bombs are there and the photon was not absorbed. D2 takes over as N grows, and the live bomb is found without being touched in almost every run. That is the paper’s point: by incorporating a detection in each cycle, the photon keeps the lower path with probability cos²(π/2N) every time. A dud would let the light through, just like an empty path.

An illustrative model of the proposal in the paper’s Fig. 4: single photons simulated one at a time, not experimental data. The probabilities are the ones the paper gives. The paper presents this chained-interferometer scheme as a proposal; the experiment it reports from Kwiat and colleagues used the polarization-and-cavity version in the next section, reaching η ≈ 0.5.

Many gentle looks

The scheme the paper treats in most detail does the same with polarization instead of paths. In this version by Kwiat and colleagues, a horizontally polarized photon passes N times through a rotator that turns its polarization by 90°/N. With nothing in the path, the small turns add up, and after N cycles the photon is vertical. If an absorbing object sits in the path, then on each cycle it effectively asks “are you here?”. The answer is almost always no, and that answer resets the photon to horizontal. The chance of surviving all N looks is [cos²(π/2N)]N , which climbs toward 1.

Redrawn after the paper’s Fig. 8: a small rotation Rx(π/n) on the photon qubit, a CNOT onto the bomb qubit, then the bomb is measured and reset, n times over.

The authors built this as a quantum circuit too. Running it through Qiskit Runtime for 1 to 10,000 cycles, they found the probability of an interaction-free measurement rising along this curve toward 1 (Fig. 8). As the paper notes, Kwiat and colleagues’ single-photon experiment reached η ≈ 0.5.

What just happened?

The paper’s answer starts with superposition. Until it is detected, the single photon travels both paths at once, and the two possibilities interfere. A bomb that could absorb the photon acts as a measurement of which path it took, and, in the paper’s words, “obtaining information about the photon’s path destroys the interference pattern.” That is why a click at the dark port can only happen when something sits in the other path, even on the runs where the photon never interacted with it.

The paper surveys how different interpretations of quantum mechanics tell this story:

Many-Worlds

The universe branches into every possible outcome. When you certify a bomb without touching it, the bomb did explode, in another branch.

Bohmian mechanics

Hidden variables guide the photon along a definite trajectory, and the bomb’s presence shapes that guidance.

A single real result

Only one outcome happens. A conclusive result arrives without direct interaction, challenging classical intuitions about what measuring means.

The authors close on why this matters beyond the puzzle. Controlling superposition and entanglement underpins quantum communication, cryptography and computing. Anton Zeilinger, an author of the interaction-free experiments the paper cites, shared the 2022 Nobel Prize in Physics for experiments with entangled photons.

Source

Vinícius Pereira Pinto, Bruno Pereira de Oliveira, Fátima Maria Mitsue Yasuoka, Philippe Wilhelm Courteille, Jarbas Caiado de Castro Neto. “Exploring Quantum Comprehension Through the Elitzur-Vaidman Bomb Testing Problem.” Brazilian Journal of Physics 53, 152 (2023). doi:10.1007/s13538-023-01366-x

Earlier work discussed in the paper: A. C. Elitzur & L. Vaidman, Found. Phys. 23, 987 (1993) · P. G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. A. Kasevich, Phys. Rev. Lett. 74, 4763 (1995) · P. G. Kwiat et al., Phys. Rev. Lett. 83, 4725 (1999).

All diagrams and animations on this page are new interactive illustrations computed from the equations in the paper. The paper’s figures and photographs are not reproduced. Photon outcomes are pseudo-random samples. Fonts: Instrument Serif, IBM Plex Sans and JetBrains Mono (SIL Open Font License).